We want to generate a normal random value from a distribution with mean $\backslash$mu $=0$ and variance $\backslash$sigma $2=1$

using the convolution method. We generate $12$ uniform$(0,1)$ values ui$,$ i $=1,2,...,12,$ and find their sum is

$4\mathrm{.}93.$

$($a$)[1$ point$]$ What standard normal random value is generated?

$($b$)[2$ points$]$ Suppose that instead of using the $12$ ui values directly, we computed wi $=1$ ui$,$ which

would themselves also be pseudo$-$random values from a uniform$(0,1)$ distribution. If we had used the

wi$$s to generate the normal value instead of the ui$$s$,$ what would its value be$?$

$($c$)[1$ point$]$ Use the answer from $($a$)$ to generate a value from a normal distribution having mean $\backslash$mu $=10$

and variance $\backslash$sigma $2=9.$

$($d$)[1$ point$]$ If we used the convolution method to generate a value from a N $(\backslash$mu $=2,\backslash$sigma $2=4)$ distribution

and obtained x $=3,$ what must have been the value of $12$

i$=1$ ui$?$

$2$